A Dantzig-Wolfe reformulation of the EndoWake model for optimal wind farm layouts

Abstract

Wind Farm Layout Optimization places turbines to maximize the expected energy under a wind rose, each turbine lowering the wind downwind. In the EndoWake model the wind field is a variable, propagated by linear inequalities, and layouts are certified by branch-and-check. On a ladder of grids it certifies the $10\times10$ grid and leaves the larger grids open after one hour. We tighten the relaxation by a convexification computed rather than derived: every pattern of turbines inside a small tile of the grid is enumerated once with its power and enters the master as a column. The device is valid because the wind field is antitone, and exact, for the cubic power curve, with a Benders cut at integer layouts. Reduced-cost fixing and propagation between overlapping tiles shrink the master before the tree search, and the tree repeats the fixing at every improving incumbent. The incumbent comes from a tile heuristic or an LP dive. On the same computer as the branch-and-check, the $12\times12$ and $14\times14$ grids close in 48 and 395 seconds. The $16\times16$ grid is certified on five seeds from random starts, one job per seed, the heuristic on a GPU, in 1413 seconds under the uniform rose and 6484 seconds under a non-uniform one. Times are medians. A pre-specified test set of 36 unseen instances with forbidden cells confirms the advantage in the axial setting. Certified means closed at the solver's gap of $10^{-4}$: the optimum lies within it plus a residual of at most 0.00042%.